Here we consider a more general class of domains.
We assume that the free surface boundary \({\mathcal {S}}\)  is given in a parametric form : \((x,y)=(u(s),v(s))\) , \(s\in R\) , where both functions are \(\Lambda \) -periodic, \(v(s)>0\)  and \(u(s)\rightarrow \pm \infty \)  when \(s\rightarrow \pm \infty \) . We assume also that the curve does not intersect itself. Here we use the same change of variables as in [1]} and [2]}.
